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\section*{Midterm}\renewcommand{\leftmark}{Midterm}\phantomsection\addcontentsline{toc}{section}{Midterm}

\textbf{1.\enspace} Find all conformal maps of the upper half
plane $\{z\lst\im(z)>0\}$ onto the lower half plane $\{z\lst\im(z)<0\}$. 

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\textbf{2.\enspace} Describe the Riemann surface of the function $f(z)=z^{2}-z$.

\bigbreak
\textbf{3.\enspace} Does the infinite product
$\displaystyle\prod^{\infty}_{n=2}\left(1+\frac{(-1)^{n}}{n}\right)$
converge? If yes, find the value of the product. Does it converge absolutely?

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\textbf{4.\enspace} Let $p(z) = a_{n} z^{n} + a_{n-1} z^{n-1} + \dots + a_{1} z + a_{0}$ be not identically 0, and assume that,
for some $k$,
\[
\|a_{k}\| > \|a_{0}\| + \|a_{1}\| + \dots + \|a_{k-1}\| + \|a_{k+1}\| + \dots + \|a_{n}\|.
\]
Prove that the polynomial $p$ has precisely $k$ roots (counted
with their multiplicities) within the disk $\{z\lst\|z\|<1\}$. 


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\textbf{5.\enspace} Find an expression, in terms of elementary functions, for $\Gamma(5 + z)\Gamma(5 - z)$.

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\textbf{6.\enspace} Find an asymptotic formula for $\displaystyle\binom{n}{k}$
(I mean an expression $F (n)$ in terms of elementary functions
whose ratio with the given binomial coefficient has limit 1 when $n\to\infty$. But I do not mind if you get a formula like this:
$\displaystyle\binom{n}{k}\asymptote[???]\cdot\left(1\pm\frac{1}{?n}+\dots\right)$.)
